University Physics with Modern Physics (14th Edition)

Published by Pearson
ISBN 10: 0321973615
ISBN 13: 978-0-32197-361-0

Chapter 41 - Quantum Mechanics II: Atomic Structure - Problems - Exercises - Page 1402: 41.17


See explanation.

Work Step by Step

We require that $\Phi(\phi)=\Phi(\phi+2 \pi)$. $\Phi(\phi+2 \pi)=e^{im_l(\phi+2 \pi)}= e^{im_l\phi} e^{im_l(2 \pi)}= \Phi(\phi) e^{im_l(2 \pi)}$ For the desired expression to be true, it is required that $e^{im_l(2 \pi)}=1$ Euler's formula states: $e^{im_l(2 \pi)}=cos(m_l(2 \pi))+i\;sin(m_l(2 \pi))$ Therefore, an inspection of the cosine graph reveals that $e^{im_l(2 \pi)}=1$ if and only if $m_l$ is an integer.
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